Problem: Solve the equation. $\dfrac{dy}{dx}=\dfrac{3}{xy^2}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\sqrt[3]{\dfrac{9x^2}{2}}+C$ (Choice B) B $y=\sqrt[3]{\dfrac{9x^2}{2}+C}$ (Choice C) C $y=\sqrt[3]{9\ln|x|+C}$ (Choice D) D $y=\sqrt[3]{9\ln|x+C|}$
Answer: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{3}{xy^2} \\\\ y^2\,dy&=\dfrac{3}{x}\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} y^2\,dy&=\dfrac{3}{x}\,dx \\\\ \int y^2\,dy&=\int \dfrac{3}{x}\,dx \\\\ \dfrac{y^3}{3}&=3\ln|x|+C_1 \\\\ y^3&=9\ln|x|+C \\\\ y&=\sqrt[3]{9\ln|x|+C} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\sqrt[3]{9\ln|x|+C}$